## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 1 (1993), 235-254.

### Testing for Additivity of a Regression Function

#### Abstract

Observations $y_{ij}$ are made at points $(x_{1i},x_{2j})$ according to the model $y_{iy}=F(x_{1i},x_{2j})+e_{ij}$, where the $e_{ij}$ are independent normals with constant variance. In order to test that $F(x_1,x_2)$ is an additive function of $x_1$ and $x_2$, a likelihood ratio test is constructed comparing $F(x_1,x_2)=\mu+Z_1 (x_1)+Z_2(x_2)$ with $F(x_1,x_2)=\mu+Z_1(x_1)+Z_2(x_2)+Z(x_1,x_2)$, where $Z_1$, $Z_2$ are Brownian motions and Z is a Brownian sheet. The ratio of Brownian sheet variance to error variance $\infty$ is chosen by maximum likelihood and the likelihood ratio test statistic W of $H_0:\infty=0$ used to test for departures from additivity. The asymptotic null distribution of W is derived, and its finite sample size behaviour is compared with two standard tests in a simulation study. The W test performs well on the five alternatives considered.

#### Article information

**Source**

Ann. Statist. Volume 21, Number 1 (1993), 235-254.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349024

**Digital Object Identifier**

doi:10.1214/aos/1176349024

**Mathematical Reviews number (MathSciNet)**

MR1212175

**Zentralblatt MATH identifier**

0771.62033

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

**Keywords**

Nonparametric regression Brownian sheet nonstandard likelihood asymptotics tests for additivity

#### Citation

Barry, Daniel. Testing for Additivity of a Regression Function. Ann. Statist. 21 (1993), no. 1, 235--254. doi:10.1214/aos/1176349024. https://projecteuclid.org/euclid.aos/1176349024